Find Dimensions to minimize cost

Question

A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $5 per square foot and the metal for the sides costs $8 per square foot. Find the dimensions that minimize cost if the box has a volume of 40 cubic feet.

Answer 1

Let x = length of a side of the bottom

     y = height

 

Given:  x²y = 40, so y = 40/(x²)

 

Minimize:

 Cost of box = 5(area of top + area of bottom) + 8(area of sides)

 

                  = 5(2x²) + 8(4xy)

 

                  = 10x² + 32xy = 10x² + 32x(40/(x²)

 

                                       = 10x² + 1280/x

 

C(x) = 10x² + 1280x^-1, where x > 0

 

       = cost of box as a function of x

 

To minimize C(x), first, set C'(x) = 0

 

     20x - 1280x^-2 = 0

 

     (20x³ - 1280)/x^{2 = 0}

 

      20x³ = 1280

 

         x³ = 64

 

         x = 4

 

When 0<x<4, C'(x) < 0. So, C(x) is decreasing

When x > 4, C'(x) > 0.  So, C(x) is increasing

 

Therefore, C(x) is minimized when x = 4 ft and y = 40/16 = 2.5 ft